I have a non-square matrix $G$ and I am looking for matrices like $F$ such that $FG=I$. I am told that it has not a unique solution. I calculated a (left?) inverse of $G$ using the formula $G^+=(G^TG)^{-1}G^T$, however I am looking for other matrices like $F$. How can I find them? Can I generate them from $G^+$?
We can assume $G^TG$ has an inverse.
P.S.: I don't know much linear algebra, please excuse me if I missed some related information, please let me know if I missed something relevant. An example of $G$ can be this:
$G=\left[\array{1 & 0.1353 & 1 \\0.3678 & 0.3678 & 1 \\ 0.1353 & 1 & 1 \\ 0.3678 & 0.3678 & 1}\right]$
If $F$ is a left inverse of $G$ and $H$ is a matrix with the same shape as $F$, such that $HG=0$, then $(G+H)F=GF+HF=I+0=I$.
So what you need are matrices $H$ such that $HG=0$; it's easier computing $H^T$ because the relation is equivalent to $G^T H^T=0$.
So you get $H^T$ by filling in the suitable number of solutions of the linear system $G^T x=0$.
Let's fix the ideas. Suppose $G$ is $m\times n$; then $H$ must be an $n\times m$ matrix and $H^T$ is $m\times n$ too. If $x_1,x_2,\dots,x_n$ are solutions of the linear homogeneous system $$ G^T x = 0, $$ and $K=[x_1~x_2~\dots~x_n]$ is the matrix with those solutions as columns, then $$ G^TK=0 $$ and so $H=K^T$ is one of the matrices you are looking for. Are you able to find nonzero solutions to $G^Tx=0$? Yes, because the rank of $G$ is $m>n$ (since $G$ has a left inverse).